I have the following problem in my Stats course, which follows Casella and Berger's book on statistical inference.
This problem is in the "Cramer-Rao lower bound" problem set. I think the solution is related to the concept of sufficient and complete statistic but the professor hasn't covered this material just yet. So, I'm trying to attack this problem only if the already-covered tools.
The whole problem set relies on the result that, given a finite sample, if we want to estimate $\theta$ using an unbiased estimator, the best one we can have when it comes to minimum variance is an estimator which attains the Cramer-Rao lower bound. Also, a necessary and sufficient condition for this attainment is that we can write the score function $S(X,\theta)$ in the following way
$S(\textbf{X},\theta) = a(\theta)(W(\textbf{X}) - \theta)$, whee $a(\theta)$ is a deterministic function of $\theta$.
Actually, Casela and Berger's book (page 341 - corollary 7.3.15) clarifies that a necessary and sufficient condition for the existence of an unbiased estimator which attains the Cramer-Rao lower bound (always supposing valid the exchange between differentiation and integration) is $S(\textbf{X},\theta) = a(\theta)(W(\textbf{X}) - g(\theta))$.
Hence, to solve the exercise I started finding the score function (let's assume that all functions involved are smooth and we are dealing with a random sample, just because we are doing this the whole course):
$S(X, \theta) = n\left[\frac{\sum\limits_{I=1}^{n} T(x_{i})}{n} - \left(- \frac{C'(\theta)}{C(\theta)} \right)\right]$
So, as far as I understand, the above result leads us to two conclusions:
1) In general, we cannot affirm that we can find and unbiased estimator which attains the Cramer-Rao lower bound when estimating $\theta$.
2) We can find and unbiased estimator that attains the Cramer-Rao lower bound for estimating the second function inside the brackets. Moreover, the estimator is $\frac{\sum\limits_{I=1}^{n} T(x_{i})}{n}$
Am I wrong up to here?
Well, since I only have these tools, I don't know how to follow on. Any ideas? Thanks so much in advance.